Nonstandard adjectives in mathematics
Ranjit Bhatnagar once propounded the notion of a "nonstandard"
adjective. This is best explained by an example. "Red" is not usually
a nonstandard adjective, because a red boat is still a boat, a red hat
is still a hat, and a red flag is still a flag. But "fake" is
typically nonstandard, because a fake diamond is not a diamond, a fake
Gucci handbag is not a Gucci handbag.

The property is not really attached to the adjective itself. Red
emeralds are not emeralds, so "red" is nonstandard when applied to
emeralds. Fake expressions of sympathy are still expressions of
sympathy, however insincere. "Toy" often goes both ways: a toy fire
engine is not a fire engine, but a toy ball is a ball and a toy dog is
a dog.

Adjectives in mathematics are rarely nonstandard. An Abelian group is
a group, a second-countable topology is a topology, an odd integer is
an integer, a partial derivative is a derivative, a well-founded order
is an order, an open set is a set, and a limit ordinal is an
ordinal.

When mathematicians want to express that a certain kind of entity is
similar to some other kind of entity, but is not actually some other
entity, they tend to use compound words. For example, a pseudometric
is not (in general) a metric. The phrase "pseudo metric" would be
misleading, because a "pseudo metric" sounds like some new kind of
metric. But there is no such term.

But there is one glaring exception. A partial function is not
(in general) a function. The containment is in the other direction:
all functions are partial functions, but not all partial functions are
functions. The terminology makes more sense if one imagines that
"function" is shorthand for "total function", but that is not usually
what people say.

If I were more quixotic, I would propose that partial functions be
called "partialfunctions" instead. Or perhaps "pseudofunctions". Or
one could go the other way and call them "normal relations", where
"normal" can be replaced by whatever adjective you
prefer—ejective relations, anyone?

I was about to write "any of these would be preferable to the current
confusion", but actually I think it probably doesn't matter very much.